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A note on the limit points associated with the generalized $$abc$$-conjecture for $$\mathbb{Z} [t]$$. (English) Zbl 0880.11026
This paper constructs examples to show that a conjectural generalization of the $$abc$$ conjecture to $$n$$ variables is best possible over function fields. Let $$K$$ be an integral domain of characteristic $$0$$. For nonzero polynomials $$A(t)\in K[t]$$ the radical of $$A$$, denoted $$\text{rad}(A)$$, is the product of all distinct irreducible factors of $$A$$. For fixed $$n\geq3$$ let $$T_n$$ denote the set of $$n$$-tuples $$(A_1(t),\dots,A_n(t))\in K[t]^n$$ such that $$A_1+\dots+A_n=0$$, such that no subsum of the $$A_i$$ equals zero, and such that $$\gcd(A_1,\dots,A_n)=1$$. The paper constructs examples showing that the limit set of the ratio $\frac{\max\{\deg A_1,\dots,\deg A_n\}} {\deg(\text{rad}(A_1\cdot\dots\cdot A_n))} ,$ as $$(A_1,\dots,A_n)$$ varies over $$T_n$$, contains the interval $$[1/n,2n-5]$$. The Masser-Oesterlé $$abc$$ conjecture asserts that, if $$n=3$$ and if $$K[t]$$ is replaced by $$\mathbb Z$$, then the lim sup of the corresponding ratio is $$\leq 1$$. This conjecture was generalized to arbitrary $$n\geq3$$ by J. Browkin and J. Brzeziński [Math. Comput. 62, 931-939 (1994; Zbl 0804.11006)], who showed that the lim sup is $$\geq 2n-5$$ over number fields, and conjectured that the lim sup was equal to $$2n-5$$.
Reviewer: P.Vojta (Berkeley)
##### MSC:
 11C08 Polynomials in number theory 11D04 Linear Diophantine equations 11A99 Elementary number theory
##### Keywords:
abc conjecture; function fields
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